Self-similar solutions for a nonlinear radiation diffusion equation

Abstract

This paper considers the hydrodynamic equations with nonlinear conduction when the internal energy and the opacity have power-law dependences in the density and in the temperature. This system models the situation in which a dense solid is brought into contact with a thermal bath. It supports self-similar solutions that depend on the surface temperature. The self-similar solution can exhibit a shock wave followed by an ablation front if the surface temperature does not increase too fast in time, but it can exhibit a heat front followed by an isothermal shock otherwise. These flows are carefully studied in order to clarify the role of the initial solid density in the energy absorption and the ablation process. Comparisons with numerical simulations show excellent agreement.

Full self-similar solutions of the subsonic radiative heat equations

“…Hammer and Rosen [12,13] proposed new solutions in this region, based on a perturbation theory. Self-similar solutions of the ablative subsonic regime in specific cases were obtained in many other works [14][15][16][17][18][19]. In particular, Garnier et.…”

“…[14] proposed a self-similar solution that includes both the ablative and shock regions, for a specific case that ensures constant density over time (which is in this particular case, selfsimilar). These solutions are widely used for obtaining a better understanding of the heat wave phenomenon, evaluating the achieved temperature in ICF experiments [3][4][5]20], or modeling hydrodynamical instabilities via linear perturbation amplification technique [21].…”

Self-similar Solution of the Subsonic Radiative Heat Equations Using a Binary Equation of State

“…This work also shows the benefit of using a solution which is composed of two different solutions for two physically different regions, as opposed to [11].…”

“…Subsonic case solutions were investigated by many authors [7][8][9][10][11]. Most of the solutions focused on the ablation region alone, since the full problem (including the shock region) is not self-similar for the general case.…”

The radiation diffusion equation: Explicit analytical solutions by improved integral-balance method

“…In general, this physical model assumes a diffusion approximation relating the local thermal flux at any point of the medium by the local gradient of the radiation energy density that is an approach known from the classical Fourier law. Following Smith The presentation of the thermal flux density  as a gradient of the 4 th power of the local temperature is an accordance with the Rosseland approximation [1,5,6] which is valid for thick, non-opaque media in absence of fluid motion [5][6][7][8][9].The diffusion equation (1) was solved for the first time by Barenblatt [10] by a selfsimilar solution and then refined by Zeldovich and Raizer [11]. As commented by Smith [5] these early attempts, especially the report of Hammer and Rosen [14] repeated the idea of the Barenblatt but met problems in defining the shape of the spatial temperature distribution profile.…”